They differ by a constant. Divide ##\frac{h^2-20h}{h-10}## out using polynomial division. Using partial fractions to begin with would have led to an easier integration problem. BTW both solutions look to have a sign error.

Try using the substitution $$u = h-10$$
After some simple algebra, you should end up with the integral $$100∫\frac{u^2-100}{u^2} du$$ which should be fairly easy to evaluate, then you can substitute your h-10 back into your solution.

Why not simply calculating it carefully? It's not allowed to give the full solution in this forum, but here's a hint:

I'd use
[tex]\frac{h^2-20h}{(h-10)^2}=\frac{(h-10)^2-100}{(h-10)^2}=1-\frac{100}{(h-10)^2}.[/tex]
This is trivial to integrate. Mathematica 9.0 under linux gets the correct result. So I wonder, why Wolfram alpha gets it wrong.